Square Triangle Picking

Author

Eric W. Weisstein
March 14, 2017

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComputationalGeometry/SquareTrianglePicking.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SquareTrianglePicking.html.

©2017 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Exact form of the area and point distributions by
Michael Trott
Jan. 27 and 31, 2005

Code

p[A_]:=4*(3-6*A-4*A*(1+A)*Pi^2+3*(-1+2*A)*(1+10*A)*Log[1-2*A]+12*A^2*(-5+Log[2*A])*Log[2*A]+24*A*(1+A)*PolyLog[2,2*A])

Area of Random Triangle: Plots

RandomTriangles[n_]:=Table[{{Hue[Random[]],Polygon[#]},Line[Append[#,First[#]]]}&@Table[Random[],{3},{2}],{n}]

l=RandomTriangles[50];
movie=Table[Graphics[
{Line[{{0,0},{1,0},{1,1},{0,1},{0,0}}],Take[l,n]},
AspectRatio->Automatic],{n,0,Length[l]}];

Show[Graphics[{Line[{{0,0},{1,0},{1,1},{0,1},{0,0}}],l},
AspectRatio->Automatic]]

Animated GIF

<<Utilities`GIF`

Export[gifs<>"sqtripic.gif",Show[#,PlotRange->{{0,1},{0,1}}]&/@movie]

Pick

TriangleArea[v_]:=Abs[Det[Append[#,1]&/@v]]/2

Exact Mean

RandomPoint

tri=RandomPoint[Rectangle[],{(10^6),3}];//Timing

Length[areas=Abs[TriangleArea/@tri]]//Timing

TimeConstrained[Length[areas=Area[Triangle[#]]&/@tri],60]//Timing

TimeConstrained[Length[areas=RegionMeasure[Triangle[#]]&/@tri],60]//Timing

bug(330535): Area, Volume, and RegionMeasure should be faster on shape primitives

Through[{Mean,Variance}[areas]]

FindRoot[p[A]==0,{A,.4}]

Plot[p[A],{A,0,1/2}]

With[{bins=100},Show[{Histogram[areas,bins],Plot[Evaluate[((1/2)/bins)Length[areas]p[A]],{A,0,(1/2)},PlotStyle->Opacity[1]]}]]

Monte Carlo Distribution

tris=Table[Random[Real],{10^5},{3},{2}];//Timing

Show[Graphics[{Line[{{0,0},{1,0},{1,1},{0,1},{0,0}}],
Red,PointSize[.02],Map[Point,Take[tris,10^3],{2}]}],
AspectRatio->Automatic,PlotRange->All]

areas=Area[Evaluate[Triangle[#]]]&/@tris;//Timing

BarChart[Length/@Split[Ceiling[300Sort[areas]]],Ticks->None,BarEdges->None,BarSpacing->-.2]

Through[{Mean,Variance}[areas]]

N[{11/144,(95/20736)},10]

Exact Distribution Function

Philip's Form

-Integrate[((Log[Abs[1-t]])/t),{t,0,x},Assumptions->0<x<1/2]

d[A_]:=4(A/3)(10-17A)-16((A^3)/3)(17-3Log[2A])Log[2A]+(2/3)(1-16A-68A^2)(1-2A)Log[1-2A]+16A^2(3+2A)(PolyLog[2,2A]-((π^2)/6))

TraditionalForm[d[A]]

TraditionalForm[expr1=FullSimplify[d[A]]]

TraditionalForm[expr2=Integrate[p[A],{A,0,A},Assumptions->0<A≤1/2]//FullSimplify]

FullSimplify[expr1-expr2,0<A<1/2]

Simplify[d'[A]-p[A]]

Limit[d[A],A->#]&/@{0,1/2}

Trott's Form

Limit[#,A->0]&/@(Derivative[#][p][A]&/@Range[0,3])

Limit[#,A->1/2]&/@(Derivative[#][p][A]&/@Range[0,3])

p[A]//TraditionalForm

Plot[p[A],{A,0,1/2},PlotRange->All,PlotStyle->Red,AxesLabel->TraditionalForm/@{A,P[A]}]

<<Utilities`Simplify`

FullSimplify[Integrate[p[A],{A,0,x},Assumptions->0<x<1/2],0<x<1/2]

LogExpand[Expand[%]]

SquareTrianglePicking /: PDF[SquareTrianglePicking,A_]:=4*(3-6*A-4*A*(1+A)*Pi^2+3*(-1+2*A)*(1+10*A)*Log[1-2*A]+12*A^2*(-5+Log[2*A])*Log[2*A]+24*A*(1+A)*PolyLog[2,2*A])
SquareTrianglePicking /:CDF[SquareTrianglePicking,x_]:=((40 x)/3)-((68 (x^2))/3)-8 (π^2) (x^2)-((16 (π^2) (x^3))/3)-(272/3) (x^3) Log[2]+16 (x^3) ((Log[2])^2)+(2/3) Log[1-2 x]-12 x Log[1-2 x]-24 (x^2) Log[1-2 x]+(272/3) (x^3) Log[1-2 x]-(272/3) (x^3) Log[x]+32 (x^3) Log[2] Log[x]+16 (x^3) ((Log[x])^2)+48 (x^2) PolyLog[2,2 x]+32 (x^3) PolyLog[2,2 x]
SquareTrianglePicking /:Domain[SquareTrianglePicking]:=Interval[{0,1/2}]

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},Plot[#[SquareTrianglePicking,A],{A,0,1/2},PlotStyle->Red,PlotRange->All,AxesLabel->TraditionalForm/@{A,#[A]}]&/@{PDF,CDF}
]],ImageSize->500]

Factor/@Collect[CDF[SquareTrianglePicking,A],{Log[1-2A],Log[A],PolyLog[2,2A]}]

ODE

Simplify[A^3 (1-2 A) pp''''[A]+A^2 pp'''[A]-4 A^2 pp''[A]+8 A pp'[A]-8 pp[A]-96 (2 A-1)/.pp:>Function[x,p[x]]]

DSolve[A^3 (1-2 A) pp''''[A]+A^2 pp'''[A]-4 A^2 pp''[A]+8 A pp'[A]-8 pp[A]-96 (2 A-1)==0,pp[A],A]

DSolve[{A^3 (1-2 A) pp''''[A]+A^2 pp'''[A]-4 A^2 pp''[A]+8 A pp'[A]-8 pp[A]-96 (2 A-1)==0,pp[1/2]==pp'[1/2]==pp''[1/2]==0},pp[A],A]

...can't figure out which direction the boundary value applies and unfortunately pick the wrong one...

pp[A_]:=A C[1]+(1/2) (-2 A+(A^2)) C[2]+A C[4] ((1/(32 A))+((Log[A])/4))+A C[3] ((1/(4 A))-2 A+2 Log[A]+2 A Log[A])+12 (-10 A-8 A Log[A]-20 (A^2) Log[A]+8 A Log[1-2 A] Log[A]+8 (A^2) Log[1-2 A] Log[A]+4 (A^2) ((Log[A])^2)-Log[-1+2 A]-8 A Log[-1+2 A]+20 (A^2) Log[-1+2 A]-8 A Log[A] Log[-1+2 A]-8 (A^2) Log[A] Log[-1+2 A]+8 A PolyLog[2,2 A]+8 (A^2) PolyLog[2,2 A])

Solve[Thread[Limit[{pp[x],pp'[x],pp''[x],pp'''[x]},x->1/2,Direction->1]==0],{C[1],C[2],C[3],C[4]}]

Solve[Thread[Limit[Simplify[{pp[x],pp'[x],pp''[x],pp'''[x]}],x->1/2,Direction->1]==0],{C[1],C[2],C[3],C[4]}]

soln=Solve[
Join[
{Limit[pp[x],x->0]==12},
Thread[Limit[Simplify[{pp[x],pp'[x],pp''[x]}],x->1/2,Direction->1]==0]
],
{C[1],C[2],C[3],C[4]}][[1]]//FullSimplify

FullSimplify[Subtract@@{pp[x]/.soln,p[x]},0<x<1/2]

Analytic Moments

Integrate[p[A],{A,0,1/2}]

(in=Integrate[A^n p[A],{A,0,1/2}])//Timing

FullSimplify[((12 (2^(1-n)) (1+(2+n) HarmonicNumber[1+n]))/((1+n) (((2+n))^3) (((3+n))^2)))]//TraditionalForm

Table[((3 (2^(3-n)) (1+(2+n) HarmonicNumber[1+n]))/((1+n) (((2+n))^3) (((3+n))^2))),{n,0,5}]

Mean

Number of pieces

V6

Length[var=Permutations[{x1,x2,x3,y1,y2,y3}]]

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

l=Function[v,Length[Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],Sequence@@({#,0,1}&/@v)]]]/@var

Sort[Tally[l]]

{Min[l],Max[l]}

var[[Flatten[Position[l,32]]]]

var[[Flatten[Position[l,4168]]]]

V7

$Version

Length[var=Permutations[{x1,x2,x3,y1,y2,y3}]]

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

(l=Function[v,Length[Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],Sequence@@({#,0,1}&/@v)]]]/@var)//Timing

Sort[Tally[l]]

Numeric

N[11/144,10]

aa={{x[1,1],x[1,2]},{x[2,1],x[2,2]},{x[3,1],x[3,2]}};

NIntegrate[area,Evaluate[Sequence@@({#,0,1}&/@Flatten[aa])],Method->QuasiMonteCarlo,PrecisionGoal->3,MaxPoints->10^6]//Timing

Integrals from v5.0

This code uses v5.0's Calculus`Integration` package

<<Calculus`Integration`

Length[i50=Calculus`Integration`Private`DebugIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]Boole[And@@(0<#<1&/@{x1,x2,x3,y1,y2,y3})],{x1},{x2},{x3},{y1},{y2},{y3}]]

Timing[i50/.HoldForm[Integrate[x__]]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo,MaxPoints->10^6]]/2

i50=(∫_0^(1/2))(∫_0^x1)(∫_0^x2)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x2^1)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x2^1)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_0^x2)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x2^1)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_0^x2)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_0^x2)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x2^1)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x2^1)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_0^x2)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x2^1)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_0^x2)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1;

Integrals from v5.2

Simplify`PiecewiseIntegrate;PiecewiseDump`$ShowIntegrals=True;

Length[i52=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]Boole[And@@(0<#<1&/@{x1,x2,x3,y1,y2,y3})],{x1,-∞,∞},{x2,-∞,∞},{x3,-∞,∞},{y1,-∞,∞},{y2,-∞,∞},{y3,-∞,∞}]]

Simplify`PiecewiseIntegrate;PiecewiseDump`$ShowIntegrals=False;

i52=(∫_0^(1/2))(∫_0^x1)(∫_0^x2)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x2^1)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x2^1)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_0^x2)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x2^1)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_0^x2)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_0^x2)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_0^x1)(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_(2 x1-x2)^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x2^1)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_x1^(2 x1))(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_0^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x2^1)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_0^(1/2))(∫_(2 x1)^1)(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_0^x2)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x1^1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_0^(-1+2 x1))(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_0^((-x1+x2)/(x2-x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_0^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x1^x2)(∫_0^1)(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_0^((-x1+x3)/(-x2+x3)))(∫_((-x2 y1+x3 y1)/(-x1+x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_(2 x1-x2)^x1)(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x2^1)(∫_0^1)(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_x1^1)(∫_x2^1)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_0^x2)(∫_0^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_0^x2)(∫_0^1)(∫_((-x2 y1+x3 y1)/(-x1+x3))^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_0^((-x1+x3)/(-x2+x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x2)/(x2-x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x1^(2 x1-x2))(∫_((-x1+x3)/(-x2+x3))^((-x1+x2)/(x2-x3)))(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_0^((-x1+x2)/(x2-x3)))(∫_0^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x2)/(x2-x3))^((-x1+x3)/(-x2+x3)))(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^((-x2 y1+x3 y1)/(-x1+x3)))(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_0^((x1-x2+x2 y1-x3 y1)/(x1-x3)))(∫_0^1)(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_(2 x1-x2)^1)(∫_((-x1+x3)/(-x2+x3))^1)(∫_((x1-x2+x2 y1-x3 y1)/(x1-x3))^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1+(∫_(1/2)^1)(∫_(-1+2 x1)^x1)(∫_x2^x1)(∫_0^1)(∫_0^1)(∫_0^((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)))(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)\[DifferentialD]y3\[DifferentialD]y2\[DifferentialD]y1\[DifferentialD]x3\[DifferentialD]x2\[DifferentialD]x1;

i50===i52

Timing[i52/.HoldForm[Integrate[x__]]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo,MaxPoints->10^6]]/2

v4.2

Most[l_List]:=Drop[l,-1]

l={};
Do[
Print[{n,(v=i[[n]]/.integrate:>IntegrateWithAssumptions)//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
]

l={(7/13824),(1/3456),((53-66 Log[2])/27648),((2-Log[2])/2304),((5-Log[64])/27648),((-29+48 Log[2])/13824),((-23+34 Log[2])/3072),((27-22 Log[2])/4608),((19-27 Log[2])/1728),((-19+72 Log[2])/13824),((-23+34 Log[2])/9216),(11/13824),((5-6 Log[2])/27648),((-13+24 Log[2])/13824),((-2+Log[8])/1728),((31-30 Log[2])/13824),((5-6 Log[2])/9216),(11/13824),((5-Log[64])/27648),((2-Log[2])/2304),((53-66 Log[2])/27648),((-29+48 Log[2])/13824),((113-96 Log[2])/13824),((-31+48 Log[2])/3456),((-37+54 Log[2])/13824),((4+Log[64])/6912),((-23+38 Log[2])/4608),(3/256)-((Log[2])/72),(77/6912),((-1+Log[4096])/1728),((23-24 Log[2])/1728),((-1+Log[4096])/1728),((23-24 Log[2])/1728),((-23+38 Log[2])/4608),((2+Log[8])/3456),((-37+54 Log[2])/13824),(3/256)-((Log[2])/72),(77/6912),((-23+34 Log[2])/9216),((-19+72 Log[2])/13824),((19-27 Log[2])/1728),((27-22 Log[2])/4608),((-23+34 Log[2])/3072),(11/13824),((5-Log[64])/27648),((2-Log[2])/2304),((53-66 Log[2])/27648),((-29+48 Log[2])/13824),(7/13824),(1/3456),((113-96 Log[2])/13824),((-31+48 Log[2])/3456),((53-66 Log[2])/27648),((2-Log[2])/2304),((5-Log[64])/27648),((-29+48 Log[2])/13824),((5-Log[64])/9216),((31-30 Log[2])/13824),((-4+Log[64])/3456),((-13+4 Log[64])/13824),((5-Log[64])/27648),(11/13824),(7/13824),(1/3456),((5-Log[64])/27648),((2-Log[2])/2304),((53-66 Log[2])/27648),((-29+48 Log[2])/13824),((-23+34 Log[2])/9216),((-19+72 Log[2])/13824),((19-27 Log[2])/1728),((27-22 Log[2])/4608),((-23+34 Log[2])/3072),(11/13824),((5-6 Log[2])/9216),((31-30 Log[2])/13824),((-2+Log[8])/1728),((-13+24 Log[2])/13824),((5-6 Log[2])/27648),(11/13824),((53-66 Log[2])/27648),((2-Log[2])/2304),((5-Log[64])/27648),((-29+48 Log[2])/13824),((113-96 Log[2])/13824),((-31+48 Log[2])/3456),((-23+38 Log[2])/4608),((4+Log[64])/6912),((-37+54 Log[2])/13824),(3/256)-((Log[2])/72),(77/6912),((-1+Log[4096])/1728),((23-24 Log[2])/1728),((-1+Log[4096])/1728),((23-24 Log[2])/1728),((-37+54 Log[2])/13824),((2+Log[8])/3456),((-23+38 Log[2])/4608),(3/256)-((Log[2])/72),(77/6912),((-23+34 Log[2])/3072),((27-22 Log[2])/4608),((19-27 Log[2])/1728),((-19+72 Log[2])/13824),((-23+34 Log[2])/9216),(11/13824),((53-66 Log[2])/27648),((2-Log[2])/2304),((5-Log[64])/27648),((-29+48 Log[2])/13824),(7/13824),(1/3456),((113-96 Log[2])/13824),((-31+48 Log[2])/3456),((5-Log[64])/27648),((2-Log[2])/2304),((53-66 Log[2])/27648),((-29+48 Log[2])/13824),((5-Log[64])/27648),((-13+4 Log[64])/13824),((-4+Log[64])/3456),((31-30 Log[2])/13824),((5-Log[64])/9216),(11/13824)};

Union[FullSimplify[l]]//Length

Sort[N[Union[FullSimplify[l]],20]]

Simplify[Plus@@l]/2

v5.0.1

l={};
Do[
Print[{n,(v=i[[n]]/.integrate:>IntegrateWithAssumptions)//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
]

v5.1.0

l={};
t=0;
Do[
Print[{n,t+=First[(v=ReleaseHold[i[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing],AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
];
Print[t]

Simplify[Plus@@l]/2

v5.2.1 Directly

Simplify[Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]Boole[And@@(0<#<1&/@{x1,x2,x3,y1,y2,y3})],{x1,-∞,∞},{x2,-∞,∞},{x3,-∞,∞},{y1,-∞,∞},{y2,-∞,∞},{y3,-∞,∞}]]/2//Timing

v5.2.1 using i50 integrals

$Version

Length[i50]

l={};
t=0;
Do[
Print[{n,t+=(dt=First[(v=ReleaseHold[i50[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing]);
{dt,v},
AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i50]}
];
Print[{t,Simplify[Plus@@l]/2}]

v5.2.1 using i52 integrals

$Version

Length[i52]

l={};
t=0;
Do[
Print[{n,t+=(dt=First[(v=ReleaseHold[i52[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing]);
{dt,v},
AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i52]}
];
Print[{t,Simplify[Plus@@l]/2}]

...crash on 9...

l={};
t=0;
Do[
Print[{n,t+=(dt=First[(v=ReleaseHold[i52[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing]);
{dt,v},
AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i52]}
];
Print[{t,Simplify[Plus@@l]/2}]

...crash on 35...

l={};
t=0;
Do[
Print[{n,t+=(dt=First[(v=ReleaseHold[i52[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing]);
{dt,v},
AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i52]}
];
Print[{t,Simplify[Plus@@l]/2}]

v6.0 Directly "(October 27, 2005)"

$Version

Simplify[Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]Boole[And@@(0<#<1&/@{x1,x2,x3,y1,y2,y3})],{x1,-∞,∞},{x2,-∞,∞},{x3,-∞,∞},{y1,-∞,∞},{y2,-∞,∞},{y3,-∞,∞}]]//Timing

...wrong...

v6.0 Directly "(October 16, 2006)": {x1,x2,x3,y1,y2,y3}

$Version

Simplify[Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1}]]/2//Timing

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1}]]

Simplify[ReleaseHold[i]/2]//Timing

l={};
t=0;
Do[
Print[{n,tt=(v=ReleaseHold[i[[n]]])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
];
Print[{t,Simplify[Total[l]]/2}]

Simplify[Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],{x1,0,1},{y1,0,1},{x2,0,1},{y2,0,1},{x3,0,1},{y3,0,1}]/2]//Timing

v6.0 Directly "(October 16, 2006)": {x1,y1,x2,y2,x3,y3}

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],{x1,0,1},{y1,0,1},{x2,0,1},{y2,0,1},{x3,0,1},{y3,0,1}]]

l={};
t=0;
Do[
Print[{n,tt=(v=ReleaseHold[i[[n]]])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
];
Print[{t,Simplify[Total[l]]/2}]

l={};
t=0;
Do[
Print[{n,tt=(v=ReleaseHold[i[[n]]])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,182,Length[i]}
];

v6.0 using v5.0 integrals (October 19, 2004)

$Version

l={};
t=0;
Do[
Print[{n,tt=(v=ReleaseHold[i50[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i50]}
];
Print[{t,Simplify[Plus@@l]/2}]

v6.0 using v5.2 integrals (October 27, 2005)

$Version

l={};
t=0;
Do[
Print[{n,tt=(v=ReleaseHold[i52[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i52]}
];
Print[{t,Simplify[Plus@@l]/2}]

v6.0 using v5.2 integrals "(October 16, 2006)"

$Version

Length[i52]

l={};
t=0;
Do[
Print[{n,tt=(v=ReleaseHold[i52[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i52]}
];
Print[{t,Simplify[Plus@@l]/2}]

FullSimplify[((4352+414 Log[2]-320 Log[8]-21 Log[64]+40 Log[4096])/55296)]/.Log[8]:>3Log[2]

...wrong...

i=i52[[{3,7,34,53,57}]]//InputForm

ReleaseHold[i/.Integrate:>IntegrateWithAssumptions]//Timing

ReleaseHold[i]//Timing

v6.0 using v5.0 integrals "(October 16, 2006)"

$Version

Length[i50]

l={};
t=0;
Do[
Print[{n,tt=(v=ReleaseHold[i50[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i50]}
];
Print[{t,Simplify[Plus@@l]/2}]

FullSimplify[((4352+414 Log[2]-320 Log[8]-21 Log[64]+40 Log[4096])/55296)]/.Log[8]:>3Log[2]

...wrong...

V6 Minimal Ordering

No Assumptions

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],Evaluate[Sequence@@({#,0,1}&/@{x1,x2,y3,y1,x3,y2})]]]

l={};
t=0;
Do[
Print[{n,tt=(v=ReleaseHold[i[[n]]])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
];
Print[{t,Simplify[Total[l]]/2}]

With Assumptions

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],Evaluate[Sequence@@({#,0,1}&/@{x1,x2,y3,y1,x3,y2})]]]

l={};
t=0;
Do[
Print[{n,tt=(v=ReleaseHold[i[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
];
Print[{t,Simplify[Total[l]]/2}]

V6.1

11/144.

(1/2)NIntegrate[Abs[(-x2)*y1+x3*y1+x1*y2-x3*y2-x1*y3+x2*y3],{y3,0,1},{y2,0,1},{y1,0,1},{x3,0,1},{x2,0,1},{x1,0,1},Method->"QuasiMonteCarlo",MaxPoints->10^7]//Timing

(1/2)IntegrateWithAssumptions[Abs[-x2*y1+x3*y1+x1*y2-x3*y2-x1*y3+x2*y3],{y3,0,1},{y2,0,1},{y1,0,1},{x3,0,1},{x2,0,1},{x1,0,1},Simplification->Simplify]//Timing

(1/2)IntegrateWithAssumptions2[Abs[-x2*y1+x3*y1+x1*y2-x3*y2-x1*y3+x2*y3],{y3,0,1},{y2,0,1},{y1,0,1},{x3,0,1},{x2,0,1},{x1,0,1},Simplification->Simplify]//Timing

?IntegrateWithAssumptions

?IntegrateWithAssumptions2

V8 IntegrateWithAssumptions

{$Version,$MachineName}

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],{x1,0,1},{y1,0,1},{x2,0,1},{y2,0,1},{x3,0,1},{y3,0,1}]]

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],{x1,0,1},{y1,0,1},{x2,0,1},{y2,0,1},{x3,0,1},{y3,0,1}]]

l={};
t=0;
Do[Print[{n,tt=(v=ReleaseHold[i[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed],ReleaseHold[i[[n]]/.Integrate:>NIntegrate]//Quiet}],{n,Length[i]}];
Print[{t,Simplify[Total[l]]/2}]

V8 Integrate

"8.0 for Linux x86 (32-bit) (January 2, 2009)"

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],{x1,0,1},{y1,0,1},{x2,0,1},{y2,0,1},{x3,0,1},{y3,0,1}]]

l={};
t=0;
Do[Print[{n,tt=(v=ReleaseHold[i[[n]]])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed],ReleaseHold[i[[n]]/.Integrate:>NIntegrate]//Quiet}],{n,Length[i]}];
Print[{t,Simplify[Total[l]]/2}]

V10 IntegrateWithAssumptions

Compute

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;
Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],{x1,0,1},{y1,0,1},{x2,0,1},{y2,0,1},{x3,0,1},{y3,0,1}]]

Off[Integrate::idiv]

l={};
t=0;
Do[Print[{n,tt=(v=ReleaseHold[i[[n]]/.Integrate:>IntegrateWithAssumptions])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed],ReleaseHold[i[[n]]/.Integrate:>NIntegrate]//Quiet}],{n,Length[i]}];

Print[{t,Simplify[Total[l]]/2}]

List

l={{1,{32.37507899999999949613993521779775619507`7.530810750113247,((563-60 (π^2)-224 Log[2]+384 ((Log[2])^2))/4608)},0.000011442021223982844`,0.000011442021266516327`},{2,{372.373391000000026451743906363844871521`8.591578552949972,((-64819+5094 (π^2)+96 (439-318 Log[2]) Log[2])/27648)},0.00005389134015744269`,0.00005389134013505865`},{3,{39.63497499999999718056642450392246246338`7.61867850187526,(1/64) (131-12 (π^2)-68 Log[2]+72 ((Log[2])^2))},0.00036493610878007843`,0.00036493613178944167`},{4,{44.24227400000000187674231710843741893768`7.666437354367204,((5 (-361+30 (π^2)))/6912)+((5 Log[2])/32)-(((Log[2])^2)/8)},0.0012916972139494498`,0.0012916972284827467`},{5,{67.70870700000000397267285734415054321289`7.851244433647321,((191-15 (π^2)+32 Log[2] (-4+Log[8]))/4608)},0.00007738372572351709`,0.00007738373872388866`},{6,{134.96948199999999928877514321357011795044`8.150835494387248,(1/432) ((70541/560)+((51 (π^2))/32)-((6927 Log[2])/35)-9 ((Log[2])^2))},0.0004350629216462104`,0.00043506291122916575`},{7,{61.0597169999999991318873071577399969101`7.806354700375854,((1437-115 (π^2)+12 Log[2] (-77+60 Log[2]))/5760)},0.0012940397670901937`,0.0012940397598513137`},{8,{20.39889900000000011459633242338895797729`7.330206640944333,((-383+553 Log[2])/16128)},0.000019245464388006188`,0.000019245463451566694`},{9,{22.19862499999999982946974341757595539093`7.3669259880272735,((513-740 Log[2])/7680)},9.256039796934346`*^-6,9.256038478345748`*^-6},{10,{46.5169289999999975293576426338404417038`7.688210948605633,((853+60 (π^2)+36 (53-160 Log[2]) Log[2])/55296)},5.275687299207317`*^-6,5.275687701466186`*^-6},{11,{90.33826700000000187174009624868631362915`7.976471668316642,((42059-39900 (π^2)+12 (44617-3360 Log[2]) Log[2])/387072)},9.538111435832342`*^-6,9.538110605573504`*^-6},{12,{18.10424799999999834199115866795182228088`7.278380403425027,((6841+320 Log[2] (-35+Log[64]))/9216)},0.000024019580247348747`,0.000024019902763160133`},{13,{41.30072100000000290265234070830047130585`7.6365575466209705,((-3875-3345 (π^2)+52224 Log[2]+1440 ((Log[2])^2))/69120)},0.000028124652211903158`,0.000028124646002042916`},{14,{42.83648800000000278487277682870626449585`7.652413770753562,((-86207-52920 (π^2)+822624 Log[2]+80640 ((Log[2])^2))/967680)},0.000451360364644686`,0.0004513603933888653`},{15,{136.56723800000000323962012771517038345337`8.155946439401522,((-29377+19950 (π^2)-255648 Log[2]+20160 ((Log[2])^2))/193536)},0.00003022769166855454`,0.00003022769144340174`},{16,{77.55021100000000444651959696784615516663`7.910182897067044,((-202-15 (π^2)+480 Log[2] (-1+Log[8]))/13824)},0.0006581038308084389`,0.0006581038553643072`},{17,{22.29361199999999954002305457834154367447`7.3687803516380175,(1/864) (-641+1050 Log[2]-180 ((Log[2])^2))},0.0003738392160490133`,0.00037381883876660976`},{18,{45.89102299999999701185515732504427433014`7.682327652349376,((1309+1115 (π^2)-32 Log[2] (544+15 Log[2]))/23040)},0.0007241901626028854`,0.000724190172327224`},{19,{74.73763800000000401269062422215938568115`7.894139281615617,((7401+4410 (π^2)-8 Log[2] (8569+840 Log[2]))/80640)},0.0022406452570484657`,0.0022406452578653644`},{20,{28.02073999999999998067323758732527494431`7.4680795136681,((535-770 Log[2])/32256)},0.00003957933311142591`,0.000039579333811457285`},{21,{26.94690399999999996794031176250427961349`7.451108788444166,((43-60 Log[2])/69120)},0.000020416220578751207`,0.000020416219641992693`},{22,{43.43939499999999753754309494979679584503`7.658483681287872,(1/432) (-(61503/64)+81 (π^2)+570 Log[2]-486 ((Log[2])^2))},0.00010951306287252135`,0.00010951358533379739`},{23,{47.1768280000000004292814992368221282959`7.694328650414039,(5665/21504)-((25 (π^2))/1152)-((5 Log[2])/32)+(((Log[2])^2)/8)},0.001007616608801859`,0.0010076166274051983`},{24,{43.48338900000000251111487159505486488342`7.658923297972683,(1/216) (-(2857/320)+((45 (π^2))/64)-((9 ((Log[2])^2))/2)+Log[64])},0.00003835701501722324`,0.000038357013792709385`},{25,{120.14473499999999717147147748619318008423`8.10030465712077,((-226213-7140 (π^2)+12 Log[2] (33353+3360 Log[2]))/1935360)},0.00005804866071124341`,0.00005804863712016897`},{26,{45.5820709999999991168806445784866809845`7.679393966538182,((-236401+19320 (π^2)+150012 Log[2]-120960 ((Log[2])^2))/967680)},0.0001493833867179905`,0.00014938338667048022`},{27,{30.17341300000000003933564585167914628983`7.500224350509071,((-3835+360 (π^2)+2004 Log[2]-2304 ((Log[2])^2))/27648)},5.815616564056561`*^-6,5.8156165707947766`*^-6},{28,{129.33433800000000246654963120818138122559`8.13231375775851,((129617-10188 (π^2)-84252 Log[2]+61056 ((Log[2])^2))/55296)},0.000017602049964161716`,0.00001760204920109978`},{29,{10.79035999999999972942532622255384922028`7.053635847620122,(107/1935360)},0.000055286871693121695`,0.000055286872307331605`},{30,{17.78229599999999877013578952755779027939`7.270587748415153,(1/34560)},0.000028935185185185186`,0.000028935184745889383`},{31,{110.25823800000000574073055759072303771973`8.063010961163565,((-795+1148 Log[2])/18432)},0.00003976580310423151`,0.000039765800842044185`},{32,{19.5530270000000001573425834067165851593`7.311813913253672,(457/1935360)},0.0002361317791005291`,0.00023613177820617736`},{33,{20.60386799999999851706888875924050807953`7.334548672161663,(1/34560)},0.000028935185185185186`,0.000028935184745889424`},{34,{72.40099399999999718602339271456003189087`7.880344441987153,((-805+1164 Log[2])/55296)},0.00003297378059491374`,0.000032973777182993645`},{35,{19.74099899999999863098310015629976987839`7.31596903979001,((-65+96 Log[2])/27648)},0.000055777247314624703`,0.00005577724589994505`},{36,{33.10896699999999981400833348743617534637`7.540545544252222,(1/864) ((833/16)-75 Log[2])},0.00008849705787512047`,0.00008849700337754884`},{37,{36.42646200000000078489392763003706932068`7.582016904766004,(1/864) (-(1043213/2240)+48 (π^2)-((402 Log[2])/35))},0.00006901793138307192`,0.00006901793171419159`},{38,{33.42591800000000290538082481361925601959`7.544683256718797,-(485/4608)+((7 (π^2))/144)-((389 Log[2])/720)},0.000028681667095575847`,0.00002868166655262598`},{39,{21.7846879999999991639469953952357172966`7.35875125763638,-(4657/69120)+((7 Log[2])/72)},0.000013730517402096853`,0.000013730516717236063`},{40,{43.39340299999999928104443824850022792816`7.658023623011746,-(213887/1935360)+((23 Log[2])/144)},0.00019565168996479865`,0.00019565169218264993`},{41,{83.77826399999999296142050297930836677551`7.94373127022352,((-4649+6708 Log[2])/55296)},0.00001141650745284795`,0.00001141650577826975`},{42,{25.35014500000000126078703033272176980972`7.4245803610720476,(2467/4608)-((149 Log[2])/96)+((9 ((Log[2])^2))/8)},0.000060718052783625964`,0.00006071905771030295`},{43,{77.24325699999999983447196427732706069946`7.908460491013932,((-36459+4480 (π^2)+59348 Log[2]-101760 ((Log[2])^2))/92160)},0.0000306845967424844`,0.00003068459329148784`},{44,{54.78167200000000036652636481449007987976`7.759235196569825,((-305323+53760 (π^2)+48 Log[2] (-7933+1680 Log[2]))/967680)},0.0000736892801230155`,0.00007368928006822045`},{45,{97.83012700000000450017978437244892120361`8.011072530593074,((-6101+480 (π^2)+48 (83-60 Log[2]) Log[2])/27648)},0.0005137369715456353`,0.0005137369602009519`},{46,{26.91290899999999908231984591111540794373`7.450560556207531,(1/864) (-2+Log[512])},0.004905468316017948`,0.004905471968952554`},{47,{271.1937730000000215113686863332986831665`8.45387962656172,((2535-192 (π^2)+16 Log[2] (-107+72 Log[2]))/9216)},0.0007432567172292643`,0.000743256710575599`},{48,{43.77934400000000181307768798433244228363`7.661869162989504,((308-24 (π^2)+(-67+48 Log[2]) Log[8])/1728)},0.0005741580356061409`,0.0005741580410027151`},{49,{39.87293900000000235195329878479242324829`7.621278161594145,((8152-840 (π^2)+201 Log[2])/15120)},0.00005654010433124986`,0.000056540105076789444`},{50,{37.90623599999999981946530169807374477386`7.59931057542731,((911-420 (π^2)+4668 Log[2])/8640)},0.00015939703660813444`,0.00015939701384183342`},{51,{522.68054099999994832614902406930923461914`8.738836245224261,(1/576) (176-10 (π^2)+(-51+20 Log[2]) Log[8])},0.00013822603910915472`,0.0001382259812269478`},{52,{152.80077099999999745705281384289264678955`8.204725458881866,-(911/1728)+((37 Log[2])/24)-((9 ((Log[2])^2))/8)},0.0008931886311983606`,0.0008931909546323559`},{53,{472.33119499999997970007825642824172973633`8.694846542677988,((1631-240 (π^2)-4797 Log[2]+8460 ((Log[2])^2))/8640)},0.00021995559496303942`,0.00021995558622551505`},{54,{174.67944499999998697603587061166763305664`8.262841716669639,((10037-1680 (π^2)+4 (3529-1680 Log[2]) Log[2])/40320)},0.00029479051150182447`,0.0002947905142889167`},{55,{46.6599069999999969127202348317950963974`7.689543782125977,((365-30 (π^2)+8 Log[2] (-29+24 Log[2]))/2304)},0.00015134581150461068`,0.00015134581310983326`},{56,{46.34095500000000100726538221351802349091`7.6865648928189625,((191-15 (π^2)+32 Log[2] (-4+Log[8]))/4608)},0.00007738372572351709`,0.00007738373872388861`},{57,{108.24854399999999543524609180167317390442`8.055021976854512,((3806+1785 (π^2)-288 Log[2] (83+35 Log[2]))/96768)},0.00011664260441572424`,0.00011664259988417254`},{58,{95.65145900000000267482391791418194770813`8.001291512115667,((-1771+170 (π^2)+800 Log[2]-960 ((Log[2])^2))/9216)},0.000012543323749330475`,0.000012543323763353207`},{59,{158.15495599999999853935150895267724990845`8.219682718951084,(1/144) (19-5 (π^2)+23 Log[2]+30 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(83-60 Log[2]) Log[2])/27648)},0.0005137369715456353`,0.0005137369602009519`},{354,{32.5650490000000019108483684249222278595`7.533351649264732,(1/864) (-2+Log[512])},0.004905468316017948`,0.00490547196895255`},{355,{41.51468899999999706551534472964704036713`7.63880170211026,$Failed},$Failed,0.0005741580410027149`},{356,{163.11420300000000338513927999883890151978`8.233091691707823,$Failed},$Failed,0.0007432567105756001`},{357,{19.50403499999999823444341018330305814743`7.310724380896283,(107/1935360)},0.000055286871693121695`,0.00005528687230733163`},{358,{24.32230200000000053250914788804948329926`7.406604589907667,(1/34560)},0.000028935185185185186`,0.000028935184745889417`},{359,{21.70870100000000135764821607153862714767`7.357233750303352,(1/288) (-(65/96)+Log[2])},0.000055777247314624703`,0.00005577724589994506`},{360,{51.52616600000000346426531905308365821838`7.7326277416131886,((-805+1164 Log[2])/55296)},0.00003297378059491374`,0.000032973777182993536`},{361,{28.72763300000000086242835095617920160294`7.478899757253219,((833-1200 Log[2])/13824)},0.00008849705787512047`,0.00008849700308439385`},{362,{214.50439000000000078216544352471828460693`8.352036098067341,((-795+1148 Log[2])/18432)},0.00003976580310423151`,0.00003976580084204416`},{363,{16.31152099999999904866854194551706314087`7.233094372855051,(457/1935360)},0.0002361317791005291`,0.00023613177820617755`},{364,{21.57272100000000136788003146648406982422`7.354504840058403,(1/34560)},0.000028935185185185186`,0.000028935184745889458`},{365,{50.51532000000000266481947619467973709106`7.7240230217440145,((911-420 (π^2)+4668 Log[2])/8640)},0.00015939703660813444`,0.000159397013841833`},{366,{57.48825999999999680767359677702188491821`7.780179077312025,((8152-840 (π^2)+201 Log[2])/15120)},0.00005654010433124986`,0.000056540105076789606`},{367,{183.08516700000001264925231225788593292236`8.283253073796535,-(911/1728)+(1/24) (37-27 Log[2]) Log[2]},0.0008931886311983606`,0.0008931909546323563`},{368,{534.15079700000001139414962381124496459961`8.748263794015632,-((π^2)/24)+((722664-5025453 Log[2]-483840 ((Log[2])^2)+2013935 Log[8])/2903040)},0.0002947905115017191`,0.0002947905142889167`},{369,{1192.37873100000001613807398825883865356445`9.097014134001743,$Failed},$Failed,0.00021995558622551513`},{370,{167.53553099999999176361598074436187744141`8.244706839761882,(1/576) (176-10 (π^2)+(-51+20 Log[2]) Log[8])},0.00013822603910915472`,0.00013822598122694772`},{371,{39.8679399999999972692421579267829656601`7.6212237092694375,-(485/4608)+((7 (π^2))/144)-((389 Log[2])/720)},0.000028681667095575847`,0.000028681666552625932`},{372,{52.8019719999999992410266713704913854599`7.743250055751421,(1/36) (-(1043213/53760)+2 (π^2)-((67 Log[2])/140))},0.00006901793138303852`,0.0000690179317141916`},{373,{28.12172599999999889064383751247078180313`7.469641885718158,(2467/4608)-((149 Log[2])/96)+((9 ((Log[2])^2))/8)},0.000060718052783625964`,0.00006071905771054156`},{374,{115.80439499999999952706275507807731628418`8.084324955296497,((-305323+53760 (π^2)-380784 Log[2]+80640 ((Log[2])^2))/967680)},0.00007368928012298543`,0.00007368927678629318`},{375,{422.25380799999999226201907731592655181885`8.646173488126413,-(4051/10240)+((7 (π^2))/144)-((31243 Log[2])/23040)-((53 ((Log[2])^2))/48)+Log[4]},0.0000306845967424918`,0.000030684593291487695`},{376,{41.42870200000000124873622553423047065735`7.637901239938568,-(213887/1935360)+((23 Log[2])/144)},0.00019565168996479865`,0.0001956516949437539`},{377,{29.37653299999999845226739125791937112808`7.488600452595805,-(4657/69120)+((7 Log[2])/72)},0.000013730517402096853`,0.000013730516717236077`},{378,{48.13968100000000305271896650083363056183`7.7031031213541,((-4649+6708 Log[2])/55296)},0.00001141650745284795`,0.00001141650577826975`},{379,{62.31152699999999811097950441762804985046`7.815168307441592,((23-32 Log[2])/9216)},0.00008889867861130108`,0.00008889866043255514`},{380,{27.98474600000000123145582620054483413696`7.467521282721442,((4127-324 (π^2)-2682 Log[2]+1944 ((Log[2])^2))/1728)},0.0024468141448254137`,0.0024468179730452095`},{381,{156.36422899999999458486854564398527145386`8.214737321052414,((619-48 (π^2)+6 Log[2] (-67+48 Log[2]))/3456)},0.0014422135911616965`,0.001442213631986736`},{382,{359.54834099999999352803570218384265899658`8.57635720248575,((-64733+5088 (π^2)+96 (439-318 Log[2]) Log[2])/27648)},0.001022582514689608`,0.0010225825022265513`},{383,{20.12394099999999852457222004886716604233`7.324312948657049,(13/387072)},0.0000335854828042328`,0.000033585482937180424`},{384,{24.27830900000000013960743672214448451996`7.405818347843316,(1/34560)},0.000028935185185185186`,0.000028935184745889397`},{385,{27.2548570000000012214513844810426235199`7.456043821005518,((-121+288 Log[2])/193536)},0.0004062623387962149`,0.0004062623376506798`},{386,{61.44265899999999902547642705030739307404`7.809069915290704,((-61925+89388 Log[2])/387072)},0.00008794274939130402`,0.00008794275031418522`},{387,{29.48751700000000042223291529808193445206`7.490238117550693,((2231-3216 Log[2])/13824)},0.00013300544843864302`,0.0001330053834179612`},{388,{176.33719300000001339867594651877880096436`8.266943836532334,((-7+Log[4096])/55296)},0.000023831130040497375`,0.0000238311261493439`},{389,{17.58032700000000048135007091332226991653`7.265626862116058,(335/387072)},0.0008654720568783069`,0.0008654721133452317`},{390,{22.38859700000000074737727118190377950668`7.370626792287447,(1/34560)},0.000028935185185185186`,0.00002893518474588944`},{391,{29.15856700000000145678313856478780508041`7.485366090014557,-(4657/69120)+((7 Log[2])/72)},0.000013730517402096853`,0.000013730516717236102`},{392,{30.03243499999999954752638586796820163727`7.498190459071395,(9325/387072)-((5 Log[2])/144)},0.000023514729234706128`,0.000023514729348075887`},{393,{29.21155900000000116278897621668875217438`7.4861546488574096,-(4691/4608)+((47 Log[2])/32)},0.00004776866964206761`,0.00004776942850194553`},{394,{47.79773399999999838883013580925762653351`7.700007221301303,((-3535+3504 Log[2]+2304 ((Log[2])^2))/27648)},0.000027179714611693494`,0.000027179714251096617`},{395,{212.32472200000000839281710796058177947998`8.347600477408404,((26965+12 Log[2] (-3641+576 Log[2]))/55296)},0.00001635957713909243`,0.000016361474990844727`},{396,{60.16385300000000313502823701128363609314`7.799935554743794,((180473+35840 (π^2)-770176 Log[2])/645120)},0.0005491982888552157`,0.0005491982948927387`},{397,{48.72559300000000348518369719386100769043`7.708357046550261,-(485/4608)+((7 (π^2))/144)-((389 Log[2])/720)},0.000028681667095575847`,0.00002868166655262602`},{398,{70.73224700000000098043528851121664047241`7.8702173681131,((-69401+40320 (π^2)-12 Log[2] (37169+3360 Log[2]))/387072)},0.000016878720108401484`,0.000016878717739404375`},{399,{1020.10092199999996864789864048361778259277`9.029243053374397,((1775-840 (π^2)+(4409+420 Log[2]) Log[4])/8064)},0.000035301705583578265`,0.000035301704907255654`},{400,{183.39911799999998720522853545844554901123`8.283997156016822,((-2363+324 (π^2)+72 (2-27 Log[2]) Log[2])/1728)},0.00032659774108833727`,0.0003266003122180419`},{401,{229.40712600000000520594767294824123382568`8.381206817404644,(1/144) (2 (π^2)-(3/8) (19+64 ((Log[2])^2)+Log[16]))},0.00030288678681921074`,0.0003028867871893295`},{402,{805.25558200000000397267285734415054321289`8.926533657294042,((2137-212 (π^2)+282 Log[2] (-3+Log[16]))/1152)},0.00016697214850129748`,0.00016697210822857645`},{403,{465.88217400000002044180291704833507537842`8.68887600668225,((-1401-280 (π^2)+6017 Log[2])/5040)},0.0012256653024146256`,0.0012256653972260645`},{404,{52.59400500000000278078005067072808742523`7.741536156600652,((911-420 (π^2)+4668 Log[2])/8640)},0.00015939703660813444`,0.00015939701384183347`},{405,{55.88650499999999965439201332628726959229`7.767906864071504,((1309+1115 (π^2)-32 Log[2] (544+15 Log[2]))/23040)},0.0007241901626028854`,0.0007241901723272234`},{406,{67.93367200000000138970790430903434753418`7.8526850033044315,((-79539+4410 (π^2)+8 Log[2] (5641+1260 Log[2]))/80640)},0.0013550068627076989`,0.001355006858224654`},{407,{142.48633799999998927887645550072193145752`8.174373138216048,((-626-(π^2)+96 Log[2] (9+Log[2]))/4608)},0.0019819984676325347`,0.0019819984128000884`},{408,{66.59187799999999413103068945929408073425`7.844021176160399,((1613-3 (π^2)-36 Log[2] (55+Log[4096]))/3456)},0.000984973169958681`,0.0009849731665938024`},{409,{156.30423799999999801002559252083301544189`8.21457066682704,((-72873+8820 (π^2)-32 Log[2] (347+420 Log[2]))/161280)},0.00014208840283901154`,0.00014208840291859604`},{410,{505.47515700000002425440470688045024871826`8.72429972910587,((81799+6690 (π^2)+96 Log[2] (-3323+1590 Log[2]))/138240)},0.0003147114561274807`,0.0003147114578734549`},{411,{33.98983299999999729834598838351666927338`7.551948944077551,((-1099+12 (245-162 Log[2]) Log[2])/1728)},0.002807900341004462`,0.002807869345457293`},{412,{44.77819199999999710826159571297466754913`7.671666467456198,-(152951/967680)+((5 (π^2))/384)+(1/288) (29-24 Log[2]) Log[2]},0.0002093106038393057`,0.00020931060785399027`},{413,{46.64790699999999645797288394533097743988`7.689432075858555,((-2857+225 (π^2)-480 Log[2] (-4+Log[8]))/69120)},0.00003835701501722242`,0.000038357013792709405`},{414,{37.6682739999999967039912007749080657959`7.596575634108083,((-82+35 Log[16])/322560)},0.00004662886061009528`,0.000046628863208936406`},{415,{34.57874300000000289401214104145765304565`7.559409115112669,((43-15 Log[16])/69120)},0.000020416220578751207`,0.000020416219641992683`},{416,{48.67360000000000042064129956997931003571`7.707893382028064,((-4843+960 (π^2)-2688 Log[2]-5760 ((Log[2])^2))/27648)},0.00004453282450125986`,0.00004453322991571308`},{417,{154.11357200000000489126250613480806350708`8.20844079979274,((1541-7140 (π^2)+71508 Log[2]+40320 ((Log[2])^2))/387072)},0.00002443649988744665`,0.000024436495348204123`},{418,{70.88022399999999834108166396617889404297`7.871124994644655,((10619-1020 (π^2)+36 Log[2] (-133+160 Log[2]))/55296)},0.000011287806291181711`,0.00001128780588732854`},{419,{49.06254100000000306636138702742755413055`7.711349950295145,((365-30 (π^2)-232 Log[2]+192 ((Log[2])^2))/2304)},0.0001513458115046045`,0.00015134581310983328`},{420,{50.29135399999999833653419045731425285339`7.722093241619035,((191-15 (π^2)-128 Log[2]+96 ((Log[2])^2))/4608)},0.00007738372572351709`,0.00007738373872388863`},{421,{70.15033599999999580631993012502789497375`7.866629668795057,(1/144) (19-5 (π^2)+23 Log[2]+30 ((Log[2])^2))},0.00005523309012492008`,0.00005523313877174643`},{422,{215.31126900000001000989868771284818649292`8.353666673880822,((3806+1785 (π^2)-23904 Log[2]-10080 ((Log[2])^2))/96768)},0.00011664260441572424`,0.0001166425998841727`},{423,{182.66623000000001297848939429968595504761`8.282258178857377,((-1771+170 (π^2)-160 Log[2] (-5+Log[64]))/9216)},0.000012543323749328932`,0.000012543323763353223`},{424,{47.54477200000000181034920387901365756989`7.697702682368454,((-3875-3345 (π^2)+52224 Log[2]+1440 ((Log[2])^2))/69120)},0.000028124652211903158`,0.00002812464600204291`},{425,{60.3218290000000010309122444596141576767`7.801074414453097,(956009/967680)-((7 (π^2))/128)-((5641 Log[2])/10080)-(((Log[2])^2)/8)},0.00023746172194838433`,0.00023746172540916086`},{426,{130.6441379999999981009750626981258392334`8.136689840994958,((2549+4 (π^2)-12 Log[2] (289+32 Log[2]))/18432)},8.140086690403928`*^-6,8.140086436691314`*^-6},{427,{41.15274399999999843657860765233635902405`7.634998711863806,((-4295+8 (π^2)+4 Log[2] (1321+288 Log[2]))/9216)},3.0826086408413603`*^-6,3.0826086427554023`*^-6},{428,{93.91072400000000186537363333627581596375`7.993315102011047,((346273-52920 (π^2)+198084 Log[2]+80640 ((Log[2])^2))/967680)},0.00001925455806789008`,0.000019254558058198785`},{429,{428.06392399999998588100424967706203460693`8.652108541562612,((-60671-4460 (π^2)+4 (55393-25440 Log[2]) Log[2])/92160)},0.000018150671153212745`,0.000018150655287980366`},{430,{88.20559099999999830288288649171590805054`7.966096027475975,((7598-10955 Log[2])/80640)},0.00005670432745287171`,0.00005670432978328798`},{431,{37.33132499999999964757080306299030780792`7.592673317885177,((513-740 Log[2])/7680)},9.256039796934346`*^-6,9.256038478345722`*^-6},{432,{63.27738000000000084810380940325558185577`7.821848402177926,(17663/27648)+((43 Log[2])/144)+((9 ((Log[2])^2))/8)-Log[4]},0.000049449196032669285`,0.000049449592685008805`}};

Unevaluated Integrals

Position[l[[All,3]],$Failed]//Flatten

bug(274325)

Failed NIntegrate

Position[l[[All,4]],$Failed]//Flatten

bug(274328)

NIntegrate issues messages

Disagreement

Subtract@@{0.000011442021223982844`,0.000011442021266516327`}

Select[l,FreeQ[#,$Failed]&&(Im[#[[-2]]]>0||Abs[Subtract@@Take[#,-2]]>1*^-6)&]

V10 Integrate with GenerateConditions -> True

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;
Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],{x1,0,1},{y1,0,1},{x2,0,1},{y2,0,1},{x3,0,1},{y3,0,1}]]

l={};
t=0;
Do[Print[{n,tt=(v=ReleaseHold[i[[n]]/.HoldPattern[Integrate[x__]]:>Integrate[x,GenerateConditions->True]])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed],ReleaseHold[i[[n]]/.Integrate:>NIntegrate]//Quiet}],{n,Length[i]}];

List

V10 Integrate

Compute

Simplify`PiecewiseThread;
PiecewiseDump`$ShowIntegrals=True;

Length[i=Integrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)],{x1,0,1},{y1,0,1},{x2,0,1},{y2,0,1},{x3,0,1},{y3,0,1}]]

l={};
t=0;
Do[Print[{n,tt=(v=ReleaseHold[i[[n]]])//Timing,t+=First[tt];AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed],ReleaseHold[i[[n]]/.Integrate:>NIntegrate]//Quiet}],{n,Length[i]}];

Moments

2

((((1/2)))^2)Integrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^2,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1}]//Timing

3

Length[i=Calculus`Integration`Private`DebugIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]^3Boole[And@@(0<#<1&/@{x1,x2,x3,y1,y2,y3})],{x1},{x2},{x3},{y1},{y2},{y3}]]

v4.2

Length[i=Calculus`Integration`Private`DebugIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]^3Boole[And@@(0<#<1&/@{x1,x2,x3,y1,y2,y3})],{x1},{x2},{x3},{y1},{y2},{y3}]]

NIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]^3,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1},Method->QuasiMonteCarlo]

Most[l_List]:=Drop[l,-1]

l={};
Do[
Print[{n,(v=i[[n]]/.HoldForm[Integrate[x__]]:>IntegrateWithAssumptions[x])//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
]

Length[l]

Position[l,_?(Im[#]=!=0&),{1},Heads->False]//Flatten

Complement[Range[124],Position[l,_?NumericQ,{1}]//Flatten]

v5

Length[i=Calculus`Integration`Private`DebugIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]^3Boole[And@@(0<#<1&/@{x1,x2,x3,y1,y2,y3})],{x1},{x2},{x3},{y1},{y2},{y3}]]

NIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]^3,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1},Method->QuasiMonteCarlo]

l={};
Do[
Print[{n,(v=i[[n]]/.HoldForm[Integrate[x__]]:>IntegrateWithAssumptions[x])//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
]

v6

l={};
Do[
Print[{n,(v=i[[n]]/.HoldForm[Integrate[x__]]:>IntegrateWithAssumptions[x])//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
]

l={};
Do[
Print[{n,(v=i[[n]]/.HoldForm[Integrate[x__]]:>IntegrateWithAssumptions[x])//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,42,Length[i]}
]

l={};
Do[
Print[{n,(v=i[[n]]/.HoldForm[Integrate[x__]]:>IntegrateWithAssumptions[x])//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,97,Length[i]}
]

l={};
Do[
Print[{n,(v=i[[n]]/.HoldForm[Integrate[x__]]:>IntegrateWithAssumptions[x])//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,120,Length[i]}
]

l

Length[l]

Select[l,Im[#]=!=0&]

FullSimplify[Plus@@l]

N[137/9000]

NIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]^3,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1},Method->QuasiMonteCarlo,PrecisionGoal->3,MaxPoints->10^7]//Timing

4

((((1/2)))^4)NIntegrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^4,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1},Method->QuasiMonteCarlo,MaxPoints->10^6]//Timing

((((1/2)))^4)Integrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^4,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1}]//Timing

N[1/2400]

5

Length[i=Calculus`Integration`Private`DebugIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]^5Boole[And@@(0<#<1&/@{x1,x2,x3,y1,y2,y3})],{x1},{x2},{x3},{y1},{y2},{y3}]]

v6

l={};
Do[
Print[{n,(v=i[[n]]/.HoldForm[Integrate[x__]]:>IntegrateWithAssumptions[x])//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
]

l={(283/337182720),(1/4214784),((319-420 Log[2])/2023096320),((64-Log[512])/7225344),((73343-103740 Log[2])/2023096320),((-12277+17920 Log[2])/337182720),((-24917+35980 Log[2])/674365440),((-9895+18144 Log[2])/67436544),((7513-10836 Log[2])/25288704),((92447-90300 Log[2])/337182720),((-24917+35980 Log[2])/96337920),(121/112394240),((319-420 Log[2])/289013760),((8903-10332 Log[2])/202309632),((-289+420 Log[2])/126443520),((-34519+50400 Log[2])/1011548160),((319-420 Log[2])/2023096320),(121/112394240),((73343-103740 Log[2])/2023096320),((64-Log[512])/7225344),((319-420 Log[2])/2023096320),((-12277+17920 Log[2])/337182720),(54979/202309632)-((Log[2])/3136),-(18593/63221760)+((Log[2])/2352),((-90431+157500 Log[2])/1011548160),((7564+315 Log[2])/126443520),((-7507+10836 Log[2])/202309632),(94463/505774080)-((Log[2])/4704),(15367/56197120),((-365+2016 Log[2])/6322176),(11209/31610880)-((Log[2])/2352),((-365+2016 Log[2])/6322176),(11209/31610880)-((Log[2])/2352),((-7507+10836 Log[2])/202309632),((60512+315 Log[256])/1011548160),((-90431+157500 Log[2])/1011548160),(94463/505774080)-((Log[2])/4704),(15367/56197120),((-24917+35980 Log[2])/96337920),((92447-90300 Log[2])/337182720),((7513-10836 Log[2])/25288704),((-9895+18144 Log[2])/67436544),((-24917+35980 Log[2])/674365440),(121/112394240),((73343-103740 Log[2])/2023096320),((64-Log[512])/7225344),((319-420 Log[2])/2023096320),((-12277+17920 Log[2])/337182720),(283/337182720),(1/4214784),(54979/202309632)-((Log[2])/3136),-(18593/63221760)+((Log[2])/2352),((319-420 Log[2])/2023096320),((64-Log[512])/7225344),((73343-103740 Log[2])/2023096320),((-12277+17920 Log[2])/337182720),((319-420 Log[2])/2023096320),((-34519+50400 Log[2])/1011548160),((-289+420 Log[2])/126443520),((8903-10332 Log[2])/202309632),((319-420 Log[2])/289013760),(121/112394240),(283/337182720),(1/4214784),((73343-103740 Log[2])/2023096320),((64-Log[512])/7225344),((319-420 Log[2])/2023096320),((-12277+17920 Log[2])/337182720),((-24917+35980 Log[2])/96337920),((92447-90300 Log[2])/337182720),((7513-10836 Log[2])/25288704),((-9895+18144 Log[2])/67436544),((-24917+35980 Log[2])/674365440),(121/112394240),((319-420 Log[2])/2023096320),((-34519+50400 Log[2])/1011548160),((-289+420 Log[2])/126443520),((8903-10332 Log[2])/202309632),((319-420 Log[2])/289013760),(121/112394240),((319-420 Log[2])/2023096320),((64-Log[512])/7225344),((73343-103740 Log[2])/2023096320),((-12277+17920 Log[2])/337182720),(54979/202309632)-((Log[2])/3136),-(18593/63221760)+((Log[2])/2352),((-7507+10836 Log[2])/202309632),((7564+315 Log[2])/126443520),((-90431+157500 Log[2])/1011548160),(94463/505774080)-((Log[2])/4704),(15367/56197120),((-365+2016 Log[2])/6322176),(11209/31610880)-((Log[2])/2352),((-365+2016 Log[2])/6322176),(11209/31610880)-((Log[2])/2352),((-90431+157500 Log[2])/1011548160),((60512+315 Log[256])/1011548160),((-7507+10836 Log[2])/202309632),(94463/505774080)-((Log[2])/4704),(15367/56197120),((-24917+35980 Log[2])/674365440),((-9895+18144 Log[2])/67436544),((7513-10836 Log[2])/25288704),((92447-90300 Log[2])/337182720),((-24917+35980 Log[2])/96337920),(121/112394240),((319-420 Log[2])/2023096320),((64-Log[512])/7225344),((73343-103740 Log[2])/2023096320),((-12277+17920 Log[2])/337182720),(283/337182720),(1/4214784),(54979/202309632)-((Log[2])/3136),-(18593/63221760)+((Log[2])/2352),((73343-103740 Log[2])/2023096320),((64-Log[512])/7225344),((319-420 Log[2])/2023096320),((-12277+17920 Log[2])/337182720),((319-420 Log[2])/289013760),((8903-10332 Log[2])/202309632),((-289+420 Log[2])/126443520),((-34519+50400 Log[2])/1011548160),((319-420 Log[2])/2023096320),(121/112394240)};

Length[l]

Select[l,Im[#]=!=0&]

N[Plus@@l]

FullSimplify[Plus@@l]

NIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]^5,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1},Method->QuasiMonteCarlo,MaxPoints->10^6,PrecisionGoal->3]//Timing

NIntegrate[Abs[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)]^5,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1},Method->QuasiMonteCarlo,MaxPoints->10^7,PrecisionGoal->4]//Timing

Timing[i/.HoldForm[Integrate[x__]]:>NIntegrate[x,Method->QuasiMonteCarlo,PrecisionGoal->2]]

6

((((1/2)))^6)Integrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^6,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1}]//Timing

8

((((1/2)))^8)Integrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^8,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1}]//Timing

Central Moments

v5.1

(i=Integrate[(A-11/144)^n p[A],{A,0,1/2}])//Timing

...crashes..

v6

(i=Integrate[(A-11/144)^n p[A],{A,0,1/2}])//Timing

Table

Table[Integrate[(A-11/144)^n p[A],{A,0,1/2}],{n,5}]

InputForm[%12]

Through[{Numerator,Denominator}[%12]]

Variance

(1/4)Integrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^2,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,1},{y2,0,1},{y3,0,1}]

1/96-(11/144)^2

Probability that a Point is Inside a Triangle

pi[x_,y_]:=(1/(2*x*(-1+y)*y))*((-1+x)^2*(-x+x^2-y+13*x*y-13*x^2*y+4*y^2-43*x*y^2+25*x^2*y^2-6*y^3+60*x*y^3-24*x^2*y^3+3*y^4-30*x*y^4+12*x^2*y^4))-(3/2)*(-1+x)^2*(-1+2*x+4*y-6*x*y-4*y^2+6*x*y^2)*Log[-1+1/x]+(3/2)*(-1+x)^2*(-1+2*y)*(-1+2*x+2*y-2*x*y-2*y^2+2*x*y^2)*Log[-1+1/y]

TraditionalForm[pi[x,y]]

8 Integrate[pi[x,y],{x,1/2,1},{y,1/2,x}]//Timing

N[11/144]

8 NIntegrate[pi[x,y],{x,1/2,1},{y,1/2,x}]//Timing

prob[x_,y_]:=Piecewise[{
{pi[x,y],1/2<=x<1 && 1/2<=y<=x},
{pi[x,1-y],1/2<=x<1&&1/2<=1-y<=x},
{pi[1-y,x],1/2<=x<1&&0<y<=1-x},
{pi[1-y,1-x],0<x<=1/2&&0<y<=x},
{pi[1-x,1-y],0<x<=1/2&&x<=y<=1/2},
{pi[1-x,y],0<x<=1/2&&x<=1-y<=1/2},
{pi[y,1-x],0<x<=1/2&&0<1-y<=x},
{pi[y,x],1/2<=x<1&&0<1-y<=1-x}
}]
ineqs=Last/@Most[(List@@prob[x,y])][[1]];

<<Graphics`InequalityGraphics`

InequalityPlot[#,{x},{y},PlotRange->{{0,1},{0,1}}]&/@ineqs

DensityPlot[prob[x,y],{x,0,1},{y,0,1},Mesh->False,ColorFunction->Hue,PlotPoints->100,AspectRatio->Automatic,FrameLabel->TraditionalForm/@{x,y}]//Timing

v5.1

NIntegrate[prob[x,y],{x,0,1},{y,0,1}]//Timing

(i=Integrate[prob[x,y],{x,0,1},{y,0,1}])//Timing

N[Last[%87]]

(i=Integrate[prob[x,y],{x,0,1},{y,0,1},GenerateConditions->True])//Timing

N[i]

Print[8NIntegrate[prob[x,y]Boole[#],{x,0,1},{y,0,1}]//Timing]&/@ineqs;

Print[8NIntegrate[prob[x,y]Boole[#],{x,0,1},{y,0,1},Method->{"SymbolicPiecewiseSubdivision","ExpandSpecialPiecewise"->{Boole}}]//Timing]&/@ineqs;

Print[{i=8Integrate[prob[x,y]Boole[#],{x,0,1},{y,0,1}],N[i]}//Timing]&/@ineqs

...hangs..

Need Simplify because otherwise we're attempting to CAD the entire Piecewise object together with the Boole, which is prohibitive.

Print[{i=8Integrate[Simplify[prob[x,y],#]Boole[#],{x,0,1},{y,0,1}],N[i]}//Timing]&/@ineqs;

Print[{i=8Integrate[Simplify[prob[x,y],#]Boole[#],{x,0,1},{y,0,1},GenerateConditions->True],N[i]}//Timing]&/@ineqs;

Print[{i=8Integrate[Simplify[prob[x,y],#]Boole[#],{x,0,1},{y,0,1},GenerateConditions->True],N[i]}//Timing]&/@ineqs;

v6

NIntegrate[prob[x,y],{x,0,1},{y,0,1}]//Timing

(i=Integrate[prob[x,y],{x,0,1},{y,0,1}])//Timing

N[i]

Print[{i=8Integrate[Simplify[prob[x,y],#]Boole[#],{x,0,1},{y,0,1}],N[i]}//Timing]&/@ineqs;

Print[{i=8Integrate[Simplify[prob[x,y],#]Boole[#],{x,0,1},{y,0,1},GenerateConditions->True],N[i]}//Timing]&/@ineqs;

FullSimplify[(1/288) (22+36 ((Log[2])^2)-1260 Log[2] Log[4]+1728 ((Log[2])^2) Log[4]-576 I π Log[4] Log[8]-3480 Log[2] Log[64]+576 I π Log[2] Log[64]+5184 ((Log[2])^2) Log[64]+786 Log[32] Log[64]-1152 Log[2] Log[32] Log[64]-9 Log[2] Log[16777216])]

Probability of an Obtuse Triangle

Monte Carlo

tris=Triangle/@Table[Random[],{10^6},{3},{2}];//Timing

{Length[tris],N[Count[tris,_?ObtuseQ]/Length[tris],10]//Timing}

{Length[tris],N[Count[tris,_?ObtuseQ]/Length[tris],10]//Timing}

Exact

N[(97/150)+(π/40),10]